L11n308

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,17,8,16 X_9,21,10,20 X_15,9,16,8 X_19,5,20,10 X_13,19,14,18 X_17,11,18,22 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 6}, {11, -2, -7, 9, -5, 3, -8, 7, -6, 4, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u v 4 + 2 u 2 v 3 -2 u v 3 - v 3 - u 2 v 2 + u v 2 - u 2 w v 2 - u w v 2 + w v 2 + v 2 + u 2 w v + 2 u w v -2 w v - u w$ (db)
Jones polynomial	$- q 7 + q 6 - q 4 + 3 q 3 -3 q 2 + 5 q -3 + 4 q -1 -2 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2 z 4 a -2 + z 4 a -4 - z 4 + a 2 z 2 -8 z 2 a -2 + 6 z 2 a -4 - z 2 a -6 - z 2 + a 2 -11 a -2 + 9 a -4 -2 a -6 + 3-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 9 a -1 + z 9 a -3 + 4 z 8 a -2 + 2 z 8 a -4 + z 8 a -6 + 3 z 8 + 2 a z 7 - z 7 a -1 -3 z 7 a -3 + z 7 a -5 + z 7 a -7 + a 2 z 6 -23 z 6 a -2 -15 z 6 a -4 -7 z 6 a -6 -14 z 6 -7 a z 5 -13 z 5 a -1 -10 z 5 a -3 -10 z 5 a -5 -6 z 5 a -7 -4 a 2 z 4 + 43 z 4 a -2 + 33 z 4 a -4 + 13 z 4 a -6 + 19 z 4 + 3 a z 3 + 26 z 3 a -1 + 37 z 3 a -3 + 23 z 3 a -5 + 9 z 3 a -7 + 3 a 2 z 2 -38 z 2 a -2 -29 z 2 a -4 -9 z 2 a -6 -15 z 2 -18 z a -1 -33 z a -3 -19 z a -5 -4 z a -7 - a 2 + 20 a -2 + 15 a -4 + 3 a -6 + 8 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

2 is the signature of L11n308. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$	$i = 3$
$r = -4$		${\mathbb Z}$	${\mathbb Z}$
$r = -3$		${\mathbb Z}^{2}$
$r = -2$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 1$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 3$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 6$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$